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Length and dimension modulo a Serre category. (English) Zbl 0878.13007
A module $$M$$ over a commutative ring $$R$$ is called $${\mathcal C}$$-simple with regard to a Serre category $${\mathcal C}$$ if for each submodule $$N$$ either $$N\in{\mathcal C}$$ or $$M/N\in{\mathcal C}$$. Using this notion of simplicity the author defines recursively $${\mathcal C}$$-dimension and $${\mathcal C}$$-length for a module, which is shown to generalize various concepts of dimensions, among others the one introduced by R. Rentschler and P. Gabriel for noetherian modules [C. R. Acad. Sci., Paris, Sér. A 265, 712-715 (1967; Zbl 0155.36201)]. In the case of $$M=R$$ the results allow certain generalizations of the classical Krull dimension.
##### MSC:
 13D05 Homological dimension and commutative rings 13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
##### Keywords:
dimensions; Krull dimension
Full Text:
##### References:
 [1] DOI: 10.1017/S0305004100075137 · Zbl 0757.13006 · doi:10.1017/S0305004100075137 [2] Rentschler K., C.R. Acad. Sci. Paris 265 pp 712– (1967) [3] DOI: 10.1093/qmath/26.1.269 · Zbl 0311.13006 · doi:10.1093/qmath/26.1.269
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